Yes you can "see your mom out". That is a perfectly natural thing to say. You can also say that you "saw/walked her to the door".
And as for your second question, any of those are fine ("she came over or came to see me or dropped by my place for a moment."). She stopped in for a visit, she came for a visit, or if you're in Louisiana, she "passed by your house"! :)
Hope that helps.